... Element Analysis. R.T. Haftka ... Finite Element Analysis. R.T. Haftka ... O(hp+1-
r)in representing the r th derivative of the field quantity. O(h2(p+1 m))i ti. t i.
1
9.5: Residuals • Because of numerical errors residual is not zero
R R K D
• Error E measure (what ( h t does d it represent?) t?) e
Error bounding • Displacement Di l error x xi e x u x ui 1 hi
x xi ui 1 hi
• Let z be the point in the element where e’=0 x
x
z
z
e ' x e ' z e ' x e ' ' s ds u ' ' s ds
• Then x
u ' ' s ds z
R.T. Haftka
x
z
u ' ' s ds
x i 1
xi
u ' ' s ds hi max u " x xi x xi 1
EML5526 Finite Element Analysis
University of Florida
4
Bounds on displacement and strain errors • Error on strains e ' x hi max u" x xi x xi1
• With some more algebra g 1 2 e x h i max 8 x i x x i 1
R.T. Haftka
u " x
EML5526 Finite Element Analysis
University of Florida
5
General features • Definitions h = approximate “characteristic length” of element: length of a linear element; length of longest line segment that fits within a plane or solid element (one option) p = degree of highest complete polynomial in the element field quantity 2m = order of the highest derivative of the field quantity in the governing differential equation • Then errors are: – O(hp+1)in representation of the field quantity – O(hp+1-r)in representing the r th derivative of the field quantity 2(p+1 m))in – O(h2(p+1-m) )i representing ti strain t i energy R.T. Haftka
EML5526 Finite Element Analysis
University of Florida
6
9.7 Multimesh extrapolation • Let L t O(hq) be b th the order d off error iin phi hi 1h2q 2 h1q 1 2 ( h1 / h2 ) q or q q q h2 h1
R.T. Haftka
1 ( h1 / h2 )
EML5526 Finite Element Analysis
University of Florida
7
Regular mesh refinement
R.T. Haftka
EML5526 Finite Element Analysis
University of Florida
8
Graphical representation • Reduced integration with hour-glass control outperforms exact or plain reduced integration.
R.T. Haftka
EML5526 Finite Element Analysis
University of Florida
9
Irregular mesh refinement
•Straight St i ht liline fitt fitted d to t three th data d t points i t R.T. Haftka